Two Spring Lecture Series in Geometric Analysis

University Of Arkansas, Fayetteville AR

Investigators

Abstract

The 35th and 36th Arkansas Spring Lectures Series in the Mathematical Sciences (SLS) will take place in 2010 and 2011 at the University of Arkansas, in Fayetteville, AR. The focus topics are the following. For SLS 2010: "Minimal Surfaces and Mean Curvature Flow." The conference will take place April 15-17, 2010. The principal speaker will be Professor William Minicozzi from Johns Hopkins University. There will be additional one-hour talks by ten invited speakers to be chosen by Professor Minicozzi, and several twenty-minute contributed talks by graduate students and recent Ph.D.'s. Professor Minicozzi will address recent progress on the study of minimal surfaces and mean curvature flow. In particular, he will speak about sharp partial regularity theorems for weak solutions of the mean curvature flow and other instances of the increased understanding of geometric flows that have accompanied recent dramatic progress on Ricci flow. Many of the tools of regularity theory that were originally invented for minimal surfaces have been widely applied to other geometric problems. In the case of mean curvature flow, this includes the monotonicity formula of Huisken, Brian White's parabolic version of Federer's dimension reducing, and Brakke's regularity theorem (which is a parabolic version of Allard's theorem for stationary varifolds). For SLS 2011: "Conformal Differential Geometry and Its Interaction with Representation Theory." The conference will take place April 7-9, 2011, and feature principal speaker Professor Michael Eastwood of the University of Adelaide. There will be additional one-hour talks by ten invited speakers to be chosen by Professor Eastwood, and several twenty-minute contributed talks by graduate students and recent Ph.D.'s. Professor Eastwood's talks will address the interplay between conformal differential geometry and Bernstein-Gelfand-Gelfand (BGG) machinery. The conformal case, important in itself, also acts as an excellent testbed for the more general theory of parabolic geometry. In particular, these talks will set the scene for a conjectured vast extension of the BGG machinery and a properly geometric theory of prolongation. Branson's Q-curvature and its prescription problem will be discussed in this context. Recent work of Oshima finds the symmetry algebras of all conformally invariant differential operators on the flat model. These lectures will discuss the curved cases, which are completely open at present. Finally, the BGG machinery can be used with complex analysis to solve reconstruction problems for integral geometry on suitable homogeneous spaces. The talks will present the current research and outlook in this direction. Mathematics conferences are vital to the development of the field. They provide unique opportunity for scientists from different subfields to meet so as to (a) summarize the most recent progress in these fields, and their interactions, (b) present new directions of research with sufficient detail to formulate and develop a list of open problems, and (c) provide an opportunity to foster new collaborations and exchange ideas toward the solution of important open questions. In the tradition of the Arkansas Spring Lectures Series, these meetings will provide unique opportunities for young researchers and graduate students to interact with prominent experts in their research areas. Public talks by Professor Daniel Rockmore (Dartmouth College) in 2010 and by Professor Eastwood (Adelaide) in 2011 will address audiences of high school students and the public at large.